Consider the set of all Lebesgue measurable functions
$f$ on $\mathbb{R}^{n}$ satisfying the inequality
\begin{equation}
|f(x)|\leq C_{1}u^{\frac{1}{1-p}}(x)\|f\|_{L_{_{p,u}}(B_{(|x|).})}
\label{N345:x3}
\end{equation}
for almost all $x\in\mathbb{R}^{n}$.
Then for all functions $f$ in this set
\begin{equation}\|Hf\|_{L_{_{p,v}}(0,\infty)}
\leq C_{2}\|f\|_{L_{p,w}(\mathbb{R}^{n})}
\label{N345:x4}
\end{equation}
where
$$
w(x)=u(x) V(|x|),\qquad x\in\mathbb{R}^{n},
$$
and
$$
C_{2}=v_{n}^{-1}pC_{1}^{1-p}.
$$

If, in addition,
\begin{equation}
\int_{B_{r_{_{2}}}\setminus B_{r_{_{1}}}}u^{\frac{1}{1-p}}(x) dx<\infty\qquad
for all \quad 0<r_{_{1}}<r_{_{2}}<\infty,
\label{N345:x5}
\end{equation}
and
\begin{equation}
\int^{1}_{0}\exp(-C^{p}_{1}\int_{B_{1}\setminus
B_{|x|}} u^{\frac{1}{1-p}}(y) dy) v(r)r^{-np}dr<\infty,
\label{N345:x6}
\end{equation}
then the constant $C_{2}$ is sharp and there exists a functions $f \in L_{p,w}(\mathbb{R}^{n})$ not equivalent to $0$,
satisfying inequality \eqref{N345:x3} and such that there is equality in inequality \eqref{N345:x4}.
Joint work with Professor V. I. Burenkov and N. Azzouz.